Given x+xy+2y^2=6 find the coordinates of all other points on this curve with slope equal to -1/3. Please show work.
Given 2y^2+xy+x=6 -------(1). Differentiate both sides to obtain 4ydy/dx+xdy/dx+y+1 = 0 or dy/dx = -(y+1)/(x+4y) = -1/3. This gives us x+4y-3(y+1)=0 -----------(2). Solve equations (1) & (2) simultaneously to obtain the two points as (2,1)&(6,-3)
To find the coordinates of all the points on the curve with a slope equal to -1/3, we need to take the derivative of the equation and set it equal to -1/3. By solving this equation, we can find the x-values of the points, and then substitute them back into the original equation to get the corresponding y-values.
Given equation: x + xy + 2y^2 = 6
Step 1: Differentiate the equation with respect to x.
We'll use implicit differentiation and treat y as a function of x.
d/dx (x + xy + 2y^2) = d/dx (6)
1 + (x * dy/dx + y) + 4y * dy/dx = 0
Step 2: Simplify the equation.
dy/dx(x + y) = -1 - 4y * dy/dx
dy/dx(x + y) + 4y*dy/dx = -1
Step 3: Factor out dy/dx.
dy/dx(x + y + 4y) = -1
dy/dx = -1 / (x + y + 4y)
Step 4: Set the derivative equal to -1/3 and solve for x and y.
-1 / (x + y + 4y) = -1/3
Cross multiply: 3(x + y + 4y) = -1
3x + 3y + 12y = -1
3x + 15y = -1
Step 5: Rearrange the equation to solve for x in terms of y.
3x = -1 - 15y
x = (-1 - 15y) / 3
x = (-1/3) - 5y
Step 6: Substitute the expression for x into the original equation to determine the corresponding y-values.
(-1/3) - 5y + (-1/3)*(-1/3) + 2y^2 = 6
Simplifying the equation:
-1/3 - 5y + 1/9 + 2y^2 = 6
Combine like terms: -15/9 - 45y/9 + 1/9 + 2y^2 = 6
-15 - 45y + 1 + 18y^2 = 54
Combine like terms: 18y^2 - 45y - 68 = 0
To continue, we can use the quadratic formula to solve for y:
y = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 18, b = -45, and c = -68.
By substituting the values and evaluating the quadratic formula, we can find the y-values for the points with a slope of -1/3. Then, substituting these y-values back into the equation x = (-1/3) - 5y, we can find the corresponding x-values.